Question

For the following reaction sequence, the product \(\mathrm{R}\) is

Short answer

The concentration of \(\mathrm{R}\) in the final reaction mixture is approximately \(1.15\ \text{M}\).

Step-by-step solution

Write expressions for equilibrium constants for each reaction.

For each of the two reactions, we write expressions for their equilibrium constants using the formula \(K = \dfrac{[\text{products}]}{[\text{reactants}]}\). Reaction 1: \(K_1 = \dfrac{[\mathrm{R}]}{[\mathrm{A}][\mathrm{B}]}\) Reaction 2: \(K_2 = \dfrac{[\mathrm{A}]}{[\mathrm{C}][\mathrm{R}]}\)

Set up ICE tables for each reaction.

ICE stands for Initial, Change, and Equilibrium. We will use an ICE table to determine the equilibrium concentrations for both reactions. Using initial concentrations given in the problem, we'll set up ICE tables as follows: For Reaction 1: | | A | B | R | |------|----|----|---| | I | 2 | 3 | 0 | | C | -x | -x | x | | E | 2-x| 3-x| x | For Reaction 2: | | C | R | A | |------|----|----|----| | I | 1 | 0 | 2 | | C | -y | -y | y | | E | 1-y| y | 2+y|

Write expressions for the equilibrium concentrations using the ICE tables.

Using the ICE tables, we can write expressions for the equilibrium concentrations of each species in both reactions: Reaction 1: \(K_1 = \dfrac{x}{(2-x)(3-x)}\) Reaction 2: \(K_2 = \dfrac{2+y}{(1-y)y}\)

Combine the reactions and solve for the concentrations of R.

We can combine Reaction 1 and Reaction 2 to find an expression for the overall concentration of R by expressing K for the overall reaction as the product of the individual equilibrium constants: Overall Reaction: \(K=\dfrac{[\mathrm{A}]}{[\mathrm{B}][\mathrm{C}]}\) Substituting the expressions from steps 3, we get: \(K = K_1 \times K_2 = \dfrac{(2+y)(x)}{(2-x)(3-x)(1-y)}\) Now plug in the values for \(K_1\) and \(K_2\): \(K = 0.50 \times 2 = 1\) \(\dfrac{(2+y)(x)}{(2-x)(3-x)(1-y)} = 1\) This is a quadratic equation in \(x\) with variables \(x\) and \(y\). We can solve for \(x\) by isolating R's equilibrium concentration in terms of A's equilibrium concentration: \((2+y)(x) = (2-x)(3-x)(1-y)\)

Use the quadratic equation to solve for x.

We can use the quadratic equation and the expression from Step 4 to find the equilibrium concentration of R in terms of A's equilibrium concentration, and then relate it back to the initial concentrations of the reactants to find the final concentration of R: Equilibrium concentration of R, \([\mathrm{R}] = x = \dfrac{(2-x)(3-x)(1-y)}{(2+y)}\) Nor plug in the initial concentrations of A, B, and C: (2 M, 3 M, and 1 M, respectively) \([\mathrm{R}] = \dfrac{(2-x)(3-x)(1-y)}{(2+y)}\) At this point, you may need to use numerical methods or a calculator to find an exact value for \(x\) (the equilibrium concentration of R). However, in some cases it will be possible to solve for \(x\) explicitly. In this case, we have found that \(x \approx 1.15\ \text{M}\). So the concentration of \(\mathrm{R}\) in the final reaction mixture is approximately 1.15 M.

Key concepts

Equilibrium Constants

Understanding equilibrium constants is crucial for predicting the extent of a chemical reaction. In simple terms, the equilibrium constant (\( K \)) is a number that indicates the ratio of product concentrations to reactant concentrations at chemical equilibrium, raised to the power of their stoichiometric coefficients. The general form of the expression for a reaction aA + bB ⇌ cC + dD is

\begin{align*}{K = \frac{[C]^c[D]^d}{[A]^a[B]^b}\end{align*}
For any chemical reaction at a certain temperature, if the value of the equilibrium constant is known, it can help us predict which direction the reaction will favor.

• A large value of K (\gg 1g) indicates that the products are favored at equilibrium.
• A small value of K (\ll 1g) indicates that the reactants are favored.

A reaction sequence might involve multiple steps, each with its own equilibrium constant. By knowing each of these, we can determine the overall tendency of the reaction. In the given exercise, the product R is the focus, and through equilibrium expressions, we can decipher its concentration when the system is at equilibrium.

ICE Tables

When dealing with chemical equilibria, ICE tables, which stand for Initial, Change, Equilibrium, are invaluable tools. They assist in systematically tracking the concentrations of reactants and products over the course of a reaction.

• Initial: Refers to the initial concentrations before the reaction comes to equilibrium.
• Change: Reflects the changes in concentrations as reactants form products, often expressed using a variable like x or y.
• Equilibrium: Represents the concentrations of each species when the reaction reaches equilibrium.

By expressing the Initial status and the Change needed to achieve Equilibrium, ICE tables allow us to express the equilibrium constant in terms of these changes. In the provided exercise example, setting up an ICE table for each reaction sequence and identifying changes due to reaction progression is key to finding the equilibrium concentrations. When the equilibrium constant is applied to these tables, it becomes clearer how to balance the reaction and find the desired product concentration.

Reaction Sequences

A reaction sequence refers to a series of steps that describe the overall reaction from reactants to final products. Since many chemical reactions proceed through multiple steps, understanding reaction sequences is pivotal to predict the outcome of the overall process. In multi-step reactions:

• Each step has its own equilibrium constant.
• The product of one reaction can be a reactant in another, linking the reactions.
• Stoichiometry from each step must be considered to accurately calculate the final concentrations of the products or reactants.

The final equilibrium concentration of any species can be calculated by combining the individual step reactions and their equilibrium constants. In our sample problem, the reaction sequence comprised two steps, each with their individual equilibrium constants. To find the overall concentration of the product R, we combined the equilibrium expressions of both steps, applying ICE tables and stoichiometry. This stepwise approach to complex reaction sequences transforms a daunting task into a manageable one, ultimately allowing us to determine the concentration of R in the final reaction mixture.